Integrand size = 14, antiderivative size = 298 \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{3/2}} \, dx=\frac {2}{3 b d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^2(c+d x)}{7 b d \sqrt {b \tan ^3(c+d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} b d \sqrt {b \tan ^3(c+d x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} b d \sqrt {b \tan ^3(c+d x)}}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} b d \sqrt {b \tan ^3(c+d x)}}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} b d \sqrt {b \tan ^3(c+d x)}} \]
2/3/b/d/(b*tan(d*x+c)^3)^(1/2)-2/7*cot(d*x+c)^2/b/d/(b*tan(d*x+c)^3)^(1/2) +1/2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(3/2)/b/d*2^(1/2)/(b*t an(d*x+c)^3)^(1/2)+1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*tan(d*x+c)^(3/2) /b/d*2^(1/2)/(b*tan(d*x+c)^3)^(1/2)-1/4*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan( d*x+c))*tan(d*x+c)^(3/2)/b/d*2^(1/2)/(b*tan(d*x+c)^3)^(1/2)+1/4*ln(1+2^(1/ 2)*tan(d*x+c)^(1/2)+tan(d*x+c))*tan(d*x+c)^(3/2)/b/d*2^(1/2)/(b*tan(d*x+c) ^3)^(1/2)
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{3/2}} \, dx=\frac {14-6 \cot ^2(c+d x)-21 \arctan \left (\sqrt [4]{-\tan ^2(c+d x)}\right ) \left (-\tan ^2(c+d x)\right )^{3/4}-21 \text {arctanh}\left (\sqrt [4]{-\tan ^2(c+d x)}\right ) \left (-\tan ^2(c+d x)\right )^{3/4}}{21 b d \sqrt {b \tan ^3(c+d x)}} \]
(14 - 6*Cot[c + d*x]^2 - 21*ArcTan[(-Tan[c + d*x]^2)^(1/4)]*(-Tan[c + d*x] ^2)^(3/4) - 21*ArcTanh[(-Tan[c + d*x]^2)^(1/4)]*(-Tan[c + d*x]^2)^(3/4))/( 21*b*d*Sqrt[b*Tan[c + d*x]^3])
Time = 0.57 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.67, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.214, Rules used = {3042, 4141, 3042, 3955, 3042, 3955, 3042, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (b \tan (c+d x)^3\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \int \frac {1}{\tan ^{\frac {9}{2}}(c+d x)}dx}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \int \frac {1}{\tan (c+d x)^{9/2}}dx}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-\int \frac {1}{\tan ^{\frac {5}{2}}(c+d x)}dx-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-\int \frac {1}{\tan (c+d x)^{5/2}}dx-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\int \frac {1}{\sqrt {\tan (c+d x)}}dx+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\int \frac {1}{\sqrt {\tan (c+d x)}}dx+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {\int \frac {1}{\sqrt {\tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \int \frac {1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {2}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )}{b \sqrt {b \tan ^3(c+d x)}}\) |
(((2*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt [2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d* x]] + Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]))/2))/d - 2/(7*d*Tan[c + d*x]^(7/2)) + 2/(3*d*Tan[c + d* x]^(3/2)))*Tan[c + d*x]^(3/2))/(b*Sqrt[b*Tan[c + d*x]^3])
3.1.34.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x] )^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2 Int[(b*Tan[c + d*x])^(n + 2), x] , x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.08 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {\tan \left (d x +c \right ) \left (21 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {7}{2}} \ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+42 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+42 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+56 b^{4} \left (\tan ^{2}\left (d x +c \right )\right )-24 b^{4}\right )}{84 d \,b^{4} {\left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )}^{\frac {3}{2}}}\) | \(233\) |
| default | \(\frac {\tan \left (d x +c \right ) \left (21 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {7}{2}} \ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+42 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+42 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+56 b^{4} \left (\tan ^{2}\left (d x +c \right )\right )-24 b^{4}\right )}{84 d \,b^{4} {\left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )}^{\frac {3}{2}}}\) | \(233\) |
1/84/d*tan(d*x+c)/b^4*(21*(b^2)^(1/4)*2^(1/2)*(b*tan(d*x+c))^(7/2)*ln((b*t an(d*x+c)+(b^2)^(1/4)*(b*tan(d*x+c))^(1/2)*2^(1/2)+(b^2)^(1/2))/(b*tan(d*x +c)-(b^2)^(1/4)*(b*tan(d*x+c))^(1/2)*2^(1/2)+(b^2)^(1/2)))+42*(b^2)^(1/4)* 2^(1/2)*(b*tan(d*x+c))^(7/2)*arctan((2^(1/2)*(b*tan(d*x+c))^(1/2)+(b^2)^(1 /4))/(b^2)^(1/4))+42*(b^2)^(1/4)*2^(1/2)*(b*tan(d*x+c))^(7/2)*arctan((2^(1 /2)*(b*tan(d*x+c))^(1/2)-(b^2)^(1/4))/(b^2)^(1/4))+56*b^4*tan(d*x+c)^2-24* b^4)/(b*tan(d*x+c)^3)^(3/2)
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{3/2}} \, dx=\frac {21 \, b^{2} d \left (-\frac {1}{b^{6} d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{2} d \left (-\frac {1}{b^{6} d^{4}}\right )^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {b \tan \left (d x + c\right )^{3}}}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{5} - 21 \, b^{2} d \left (-\frac {1}{b^{6} d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{2} d \left (-\frac {1}{b^{6} d^{4}}\right )^{\frac {1}{4}} \tan \left (d x + c\right ) - \sqrt {b \tan \left (d x + c\right )^{3}}}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{5} + 21 i \, b^{2} d \left (-\frac {1}{b^{6} d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{2} d \left (-\frac {1}{b^{6} d^{4}}\right )^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {b \tan \left (d x + c\right )^{3}}}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{5} - 21 i \, b^{2} d \left (-\frac {1}{b^{6} d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{2} d \left (-\frac {1}{b^{6} d^{4}}\right )^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {b \tan \left (d x + c\right )^{3}}}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{5} + 4 \, \sqrt {b \tan \left (d x + c\right )^{3}} {\left (7 \, \tan \left (d x + c\right )^{2} - 3\right )}}{42 \, b^{2} d \tan \left (d x + c\right )^{5}} \]
1/42*(21*b^2*d*(-1/(b^6*d^4))^(1/4)*log((b^2*d*(-1/(b^6*d^4))^(1/4)*tan(d* x + c) + sqrt(b*tan(d*x + c)^3))/tan(d*x + c))*tan(d*x + c)^5 - 21*b^2*d*( -1/(b^6*d^4))^(1/4)*log(-(b^2*d*(-1/(b^6*d^4))^(1/4)*tan(d*x + c) - sqrt(b *tan(d*x + c)^3))/tan(d*x + c))*tan(d*x + c)^5 + 21*I*b^2*d*(-1/(b^6*d^4)) ^(1/4)*log((I*b^2*d*(-1/(b^6*d^4))^(1/4)*tan(d*x + c) + sqrt(b*tan(d*x + c )^3))/tan(d*x + c))*tan(d*x + c)^5 - 21*I*b^2*d*(-1/(b^6*d^4))^(1/4)*log(( -I*b^2*d*(-1/(b^6*d^4))^(1/4)*tan(d*x + c) + sqrt(b*tan(d*x + c)^3))/tan(d *x + c))*tan(d*x + c)^5 + 4*sqrt(b*tan(d*x + c)^3)*(7*tan(d*x + c)^2 - 3)) /(b^2*d*tan(d*x + c)^5)
\[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{\left (b \tan ^{3}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.49 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{3/2}} \, dx=\frac {\frac {21 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{b^{\frac {3}{2}}} + \frac {8 \, {\left (21 \, \sqrt {\tan \left (d x + c\right )} + \frac {7}{\tan \left (d x + c\right )^{\frac {3}{2}}} - \frac {3}{\tan \left (d x + c\right )^{\frac {7}{2}}}\right )}}{b^{\frac {3}{2}}} - \frac {168 \, \sqrt {\tan \left (d x + c\right )}}{b^{\frac {3}{2}}}}{84 \, d} \]
1/84*(21*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqrt(2) *log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*log(-sqrt(2) *sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/b^(3/2) + 8*(21*sqrt(tan(d*x + c) ) + 7/tan(d*x + c)^(3/2) - 3/tan(d*x + c)^(7/2))/b^(3/2) - 168*sqrt(tan(d* x + c))/b^(3/2))/d
Time = 0.62 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{3/2}} \, dx=\frac {1}{84} \, b^{4} {\left (\frac {42 \, \sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{6} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {42 \, \sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{6} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {21 \, \sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{6} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} - \frac {21 \, \sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{6} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {8 \, {\left (7 \, b^{2} \tan \left (d x + c\right )^{2} - 3 \, b^{2}\right )}}{\sqrt {b \tan \left (d x + c\right )} b^{7} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right ) \tan \left (d x + c\right )^{3}}\right )} \]
1/84*b^4*(42*sqrt(2)*sqrt(abs(b))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) + 2*sqrt(b*tan(d*x + c)))/sqrt(abs(b)))/(b^6*d*sgn(tan(d*x + c))) + 42*sq rt(2)*sqrt(abs(b))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) - 2*sqrt(b*ta n(d*x + c)))/sqrt(abs(b)))/(b^6*d*sgn(tan(d*x + c))) + 21*sqrt(2)*sqrt(abs (b))*log(b*tan(d*x + c) + sqrt(2)*sqrt(b*tan(d*x + c))*sqrt(abs(b)) + abs( b))/(b^6*d*sgn(tan(d*x + c))) - 21*sqrt(2)*sqrt(abs(b))*log(b*tan(d*x + c) - sqrt(2)*sqrt(b*tan(d*x + c))*sqrt(abs(b)) + abs(b))/(b^6*d*sgn(tan(d*x + c))) + 8*(7*b^2*tan(d*x + c)^2 - 3*b^2)/(sqrt(b*tan(d*x + c))*b^7*d*sgn( tan(d*x + c))*tan(d*x + c)^3))
Timed out. \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}^{3/2}} \,d x \]